Jacobi, an only child, was born in Leytonstone, London, England, the son of Daisy Gertrude (née Masters), a secretary who worked in a drapery store in Leyton High Road, and Alfred George Jacobi, who ran a sweet shop and was a tobacconist in Chingford.[6] His great-grandfather on his father's side had emigrated to England from Germany during the 19th century.[7] His family was working class.[8] Jacobi describes his childhood as happy. In his teens he went to the Leyton Sixth Form College and became an integral part of the drama club, The Players of Leyton.

While in the sixth-form, he starred in a production of Hamlet, which was taken to the Edinburgh Festival Fringe and very well regarded.[9] At 18 he won a scholarship to the University of Cambridge, where he read history at St John's College and earned his degree. Younger members of the university at the time included Ian McKellen (who had a crush on him – "a passion that was undeclared and unrequited", as McKellen relates it)[10] and Trevor Nunn. During his studies at Cambridge, Jacobi played many parts including Hamlet, which was taken on a tour to Switzerland, where he met Richard Burton. As a result of his performance of Edward II at Cambridge, Jacobi was invited to become a member of the Birmingham Repertory Theatre immediately upon his graduation in 1960.

Jacobi's talent was recognised by Laurence Olivier, who invited the young actor back to London to become one of the founding members of the new National Theatre, even though at the time Jacobi was relatively unknown. He played Laertes in the National Theatre's inaugural production of Hamlet opposite Peter O'Toole in 1963. Olivier cast him as Cassio in the successful National Theatre stage production of Othello, a role that Jacobi repeated in the 1965 film version. He played Andrei in the NT production and film of Three Sisters (1970), both featuring Olivier. On 27 July 1965, Jacobi played Brindsley Miller in the first production of Peter Shaffer's Black Comedy. It was presented by the National Theatre at Chichester and subsequently in London.

Jacobi was increasingly busy with stage and screen acting, but his big breakthrough came in 1976 when he played the title role in the BBC's series I, Claudius. He cemented his reputation with his performance as the stammering, twitching Emperor Claudius, winning much praise. In 1979, thanks to his international popularity, he took Hamlet on a theatrical world tour through England, Egypt, Greece, Sweden, Australia, Japan and China, playing Prince Hamlet. He was invited to perform the role at Kronborg Castle, Denmark, known as Elsinore Castle, the setting of the play. In 1978 he appeared in the BBC Television Shakespeare production of Richard II, with Sir John Gielgud and Dame Wendy Hiller.

In 2001, Jacobi won an Emmy Award by mocking his Shakespearean background in the television sitcom Frasier episode "The Show Must Go Off", in which he played the world's worst Shakespearean actor: the hammy, loud, untalented Jackson Hedley. This was his first guest appearance on an American television programme.

In 2003, he was involved with Scream of the Shalka, a webcast based on the science fiction series Doctor Who. He played the voice of the Doctor's nemesis the Master alongside Richard E. Grant as the Doctor. In the same year, he also appeared in Deadline, an audio drama also based on Doctor Who. Therein he played Martin Bannister, an ageing writer who makes up stories about "the Doctor", a character who travels in time and space, the premise being that the series had never made it on to television. Jacobi later followed this up with an appearance in the Doctor Who episode "Utopia" (June 2007); he appears as the kindly Professor Yana, who by the end of the episode is revealed to actually be the Master. Jacobi admitted to Doctor Who Confidential he had always wanted to be on the show: "One of my ambitions since the '60s has been to take part in a Doctor Who. The other one is Coronation Street. So I've cracked Doctor Who now. I'm still waiting for Corrie."[11]

In 2013, Jacobi starred alongside Ian McKellen in the ITV sitcom Vicious as Stuart Bixby, the partner to Freddie Thornhill, played by McKellen. On 23 August 2013 the show was renewed for a six-episode second series which began airing in June 2015.[16]

In 2011, Jacobi accepted a role in the film Anonymous, about the Oxfordian theory, starring Rhys Ifans and Vanessa Redgrave. In the film Jacobi narrates the Prologue and Epilogue, set in modern-day New York, while the film proper is set in Elizabethan England. Jacobi allows that making the film was "a very risky thing to do", and imagines that "the orthodox Stratfordians are going to be apoplectic with rage".[22]

Jacobi is openly gay. In March 2006, four months after civil partnerships were introduced in the United Kingdom, Jacobi registered his civil partnership with Richard Clifford.[23] They live in Primrose Hill, north London.[24] He was a grand marshall of the 46th New York City Gay Pride March in 2015.[25]